Finite Group Databasebeta

← #11736915

Subgroups — #11736915

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90234C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic
3C7×#435C211yesmaximal, characteristic
4#89909C211yesmaximal, characteristic, complemented
5#89893C211yesmaximal, characteristic, complemented
6#89888C211yesmaximal, characteristic, complemented
7C2×#6391C211yesmaximal, characteristic, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89601C211yesmaximal, characteristic, complemented
10#89495C211yesmaximal, characteristic, complemented
11#89494C211yesmaximal, characteristic, complemented
12#89493C211yesmaximal, characteristic, complemented
13#89493C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6498—21no—
20D4×Dic7—21no—
21#6489C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25#6487C2211yescharacteristic
26#6487C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30#6453—21no—
31#6452—21no—
32#6442—21no—
33#6441—21no—
34#6441—21no—
35#6440—21no—
36#6433C2211yescharacteristic
37#6432C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42C7×C22⋊C8C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6392C2211yescharacteristic
45#6392C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3344—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7—21no—
64C22×Dic7—21no—
65C22×Dic7—21no—
66C2×Dic14—21no—
67C2×Dic14C2311yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×M16—22no—
70C2×C7×C8—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4D411yescharacteristic
76C7×C22⋊C4D411yescharacteristic
77C7×C22⋊C4D411yescharacteristic
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105Dic28—22no—
106#554—71no—
107#552—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#483—71no—
112#483—71no—
113#479—71no—
114#475—71no—
115#470—71no—
116#467—71no—
117#441—71no—
118#435D711yescharacteristic
119#428—71no—
120C2×#106—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C7×D4—22no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—42no—
142C2×Dic7—81no—
143C2×Dic7—81no—
144Dic14—21no—
145Dic14—21no—
146Dic14—21no—
147Dic14—21no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—21no—
151C7×C8—21no—
152C7×C8#4112yes—
153C2×Q16—71no—
154C2×M16C2×D711yescharacteristic
155C22×C8C2×D711yescharacteristic
156C4⋊Q8—71no—
157#126—141no—
158#125—71no—
159#125—141no—
160#125—141no—
161#123—141no—
162C22⋊Q8—71no—
163C22⋊Q8—71no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×D4—141no—
171C4×D4C2×D711yescharacteristic
172#117—71no—
173#117—141no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177#107—71no—
178#107—71no—
179#107—71no—
180#106—71no—
181#106—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186C4⋊C8C2×D711yescharacteristic
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193C22⋊C8C2×D711yescharacteristic
194C22⋊C8C2×D711yescharacteristic
195#97C2×D711yescharacteristic
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#13712yes—
202C22×C7C22×D411yescharacteristic
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7C2×#4111yescharacteristic
208C4×C72−1+411yescharacteristic
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C2×Q8—71no—
218C2×Q8—71no—
219C2×Q8—141no—
220C2×D4D2811yescharacteristic
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—141no—
224C22×C4C22×D711yescharacteristic
225C22×C4C22×D711yescharacteristic
226Q16—142no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C2×C8C22×D711yescharacteristic
232C2×C8C22×D711yescharacteristic
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4D2811yescharacteristic
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4D2811yescharacteristic
258C22⋊C4D2811yescharacteristic
259C42—141no—
260C42—141no—
261C42—141no—
262C42C22×D711yescharacteristic
263C2×C7—22no—
264C2×C7—41no—
265C2×C7#54711yescharacteristic
266C2×C7C2×#13711yescharacteristic
267C2×C7#57711yescharacteristic
268C23C2×D2811yescharacteristic
269C23C2×D2811yescharacteristic
270Q8—141no—
271Q8—141no—
272Q8—141no—
273Q8—141no—
274Q8—561no—
275D4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—282no—
289C2×C4—561no—
290C2×C4—561no—
291C2×C4C2×D2811yescharacteristic
292C2×C4C2×D2811yescharacteristic
293C2×C4C2×D2811yescharacteristic
294C2×C4C2×D2811yescharacteristic
295C2×C4C23×D711yescharacteristic
296C8—21no—
297C8—21no—
298C8—21no—
299C8#115912yes—
300C7#334411yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22#647212yes—
307C22C22×D2811yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4—281no—
320C4C2×#115911yescharacteristic
321C4#655411yescharacteristic
322C2—22no—
323C2—41no—
324C2#9024611yescharacteristic
325C2C2×#647211yescharacteristic
326C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radicalC2×C7×D4D4
hypercenter#435D7
hypocenterC7#3344
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #89909C2semidirect
5. #89893C2semidirect
6. #89888C2semidirect
7. C2×#6391C2semidirect
8. #89623C2semidirect
9. #89601C2semidirect
10. #89495C2semidirect
11. #89494C2semidirect
12. #89493C2semidirect
13. #89493C2semidirect
14. #89488C2semidirect
15. #89474C2semidirect
300. C7#3344semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#902342
#902322
C7×#4352
#899092
#898932
#898882
C2×#63912
#896232
#896012
#894952
#894942
#894932
#894882
#894742
#33447